The residual trick

Every edge (u, v, c) becomes TWO arcs: forward with capacity c, backward with capacity 0. Pushing f units along u→v: cap(u→v) -= f, cap(v→u) += f. The backward arc is the UNDO button — later paths may cancel earlier flow, which is exactly why greedy pushing without back-edges is wrong.

Termination: Edmonds–Karp (BFS shortest augmenting path) guarantees O(V·E) augmentations — the bound that makes flow polynomial.